On Low-High Orders of Directed Graphs: Incremental Algorithms and Applications

A flow graph $G=(V,E,s)$ is a directed graph with a distinguished start vertex $s$. The dominator tree $D$ of $G$ is a tree rooted at $s$, such that a vertex $v$ is an ancestor of a vertex $w$ if and only if all paths from $s$ to $w$ include $v$. The dominator tree is a central tool in program optimization and code generation and has many applications in other diverse areas including constraint programming, circuit testing, biology, and in algorithms for graph connectivity problems. A low-high order of $G$ is a preorder $δ$ of $D$ that certifies the correctness of $D$ and has further applications in connectivity and path-determination problems. In this paper, we first consider how to maintain efficiently a low-high order of a flow graph incrementally under edge insertions. We present algorithms that run in $O(mn)$ total time for a sequence of $m$ edge insertions in an initially empty flow graph with $n$ vertices.These immediately provide the first incremental certifying algorithms for maintaining the dominator tree in $O(mn)$ total time, and also imply incremental algorithms for other problems. Hence, we provide a substantial improvement over the $O(m^2)$ simple-minded algorithms, which recompute the solution from scratch after each edge insertion. We also show how to apply low-high orders to obtain a linear-time $2$-approximation algorithm for the smallest $2$-vertex-connected spanning subgraph problem (2VCSS). Finally, we present efficient implementations of our new algorithms for the incremental low-high and 2VCSS problems and conduct an extensive experimental study on real-world graphs taken from a variety of application areas. The experimental results show that our algorithms perform very well in practice.